Stability region analysis using sum of squares programming

We propose using (bilinear) sum of squares programming for enlarging a provable region of attraction of polynomial systems using polynomial Lyapunov functions. In order to keep the number of decision variables low, we propose the use of composite Lyapunov functions. We also present a method of finding outer bounds of attractive invariant sets for polynomial systems. Excellent results for some examples were obtained using the proposed methods and the PENBMI solver

[1]  Y. H. Ku,et al.  Stability study of a third-order servomechanism with multiplicative feedback control , 1958, Transactions of the American Institute of Electrical Engineers, Part II: Applications and Industry.

[2]  Verne C. Fryklund,et al.  What systems analysis? , 1981, Nature.

[3]  E. Davison,et al.  A computational method for determining quadratic lyapunov functions for non-linear systems , 1971 .

[4]  C. A. Desoer,et al.  Nonlinear Systems Analysis , 1978 .

[5]  A. Vicino,et al.  On the estimation of asymptotic stability regions: State of the art and new proposals , 1985 .

[6]  M-F Roy,et al.  Géométrie algébrique réelle , 1987 .

[7]  J. Thorp,et al.  Stability regions of nonlinear dynamical systems: a constructive methodology , 1989 .

[8]  M. Vidyasagar,et al.  Nonlinear systems analysis (2nd ed.) , 1993 .

[9]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[10]  Kim-Chuan Toh,et al.  SDPT3 -- A Matlab Software Package for Semidefinite Programming , 1996 .

[11]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[12]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[13]  Pablo A. Parrilo,et al.  Introducing SOSTOOLS: a general purpose sum of squares programming solver , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[14]  Peter J Seiler,et al.  SOSTOOLS: Sum of squares optimization toolbox for MATLAB , 2002 .

[15]  Pablo A. Parrilo,et al.  Semidefinite programming relaxations for semialgebraic problems , 2003, Math. Program..

[16]  Z. Jarvis-Wloszek,et al.  Lyapunov Based Analysis and Controller Synthesis for Polynomial Systems using Sum-of-Squares Optimization , 2003 .

[17]  Johan Efberg,et al.  YALMIP : A toolbox for modeling and optimization in MATLAB , 2004 .

[18]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[19]  A. Papachristodoulou.  Scalable analysis of nonlinear systems using convex optimization , 2005 .